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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2512.19672 |
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| _version_ | 1866912783146156032 |
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| author | Blanc-Renaudie, Arthur Nachmias, Asaf |
| author_facet | Blanc-Renaudie, Arthur Nachmias, Asaf |
| contents | We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a large enough constant for the nearest neighbor model, or any fixed $d>6$ for spread-out models. We prove that there exist constants $\mathbf{C},\mathbf{C}'$ depending only on the dimension and the spread-out parameter such that for any $λ\in \mathbb{R}$ if the edge probability is $p_c(\mathbb{Z}^d)+\mathbf{C} λn^{-d/3} + o(n^{-d/3})$, then the joint distribution of the largest clusters normalized by $\mathbf{C}' n^{-2d/3}$ converges as $n\to \infty$ to the ordered lengths of excursions above past minimum of an inhomogeneous Brownian motion started at $0$ with drift $λ-t$ at time $t\in[0,\infty)$. This canonical limit was identified by Aldous in the context of critical Erdős--Rényi graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19672 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Critical percolation on the discrete torus in high dimensions Blanc-Renaudie, Arthur Nachmias, Asaf Probability We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a large enough constant for the nearest neighbor model, or any fixed $d>6$ for spread-out models. We prove that there exist constants $\mathbf{C},\mathbf{C}'$ depending only on the dimension and the spread-out parameter such that for any $λ\in \mathbb{R}$ if the edge probability is $p_c(\mathbb{Z}^d)+\mathbf{C} λn^{-d/3} + o(n^{-d/3})$, then the joint distribution of the largest clusters normalized by $\mathbf{C}' n^{-2d/3}$ converges as $n\to \infty$ to the ordered lengths of excursions above past minimum of an inhomogeneous Brownian motion started at $0$ with drift $λ-t$ at time $t\in[0,\infty)$. This canonical limit was identified by Aldous in the context of critical Erdős--Rényi graphs. |
| title | Critical percolation on the discrete torus in high dimensions |
| topic | Probability |
| url | https://arxiv.org/abs/2512.19672 |